Inquantum information theory,quantum state purification refers to the process of representing amixed state as apure quantum state of higher-dimensionalHilbert space. The purification allows the original mixed state to be recovered by taking thepartial trace over the additional degrees of freedom. The purification is not unique, the different purifications that can lead to the same mixed states are limited by theSchrödinger–HJW theorem.

Purification is used in algorithms such asentanglement distillation,magic state distillation andalgorithmic cooling.

Let${\displaystyle {\mathcal{H}}_S}$ be afinite-dimensionalcomplexHilbert space, and consider a generic (possiblymixed)quantum state${\displaystyle \rho }$ defined on${\displaystyle {\mathcal{H}}_S}$ and admitting a decomposition of the form$${\displaystyle \rho =\sum _i{p}_i|{\varphi }_i⟩⟨{\varphi }_i|}$$for a collection of (not necessarily mutually orthogonal) states${\displaystyle |{\varphi }_i⟩\in {\mathcal{H}}_S}$ and coefficients${\displaystyle p_i\ge 0}$ such that$\sum _i{p}_i=1$. Note that any quantum state can be written in such a way for some${\displaystyle \{|{\varphi }_i⟩{\}}_i}$ and${\displaystyle \{p_i{\}}_i}$.^[1]

Any such${\displaystyle \rho }$ can bepurified, that is, represented as thepartial trace of apure state defined in a larger Hilbert space. More precisely, it is always possible to find a (finite-dimensional) Hilbert space${\displaystyle {\mathcal{H}}_A}$ and a pure state${\displaystyle |{\mathrm{\Psi }}_{SA}⟩\in {\mathcal{H}}_S\otimes {\mathcal{H}}_A}$ such that${\displaystyle \rho ={\mathrm{Tr}}_A(|{\mathrm{\Psi }}_{SA}⟩⟨{\mathrm{\Psi }}_{SA}|)}$. Furthermore, the states${\displaystyle |{\mathrm{\Psi }}_{SA}⟩}$ satisfying this are all and only those of the form$${\displaystyle |{\mathrm{\Psi }}_{SA}⟩=\sum _i\sqrt{p_i}|{\varphi }_i⟩\otimes |a_i⟩}$$for some orthonormal basis${\displaystyle \{|a_i⟩{\}}_i\subset {\mathcal{H}}_A}$. The state${\displaystyle |{\mathrm{\Psi }}_{SA}⟩}$ is then referred to as the "purification of${\displaystyle \rho }$". Since the auxiliary space and the basis can be chosen arbitrarily, the purification of a mixed state is not unique; in fact, there are infinitely many purifications of a given mixed state.^[2] Because all of them admit a decomposition in the form given above, given any pair of purifications${\displaystyle |\mathrm{\Psi }⟩,|{\mathrm{\Psi }}^{\prime }⟩\in {\mathcal{H}}_S\otimes {\mathcal{H}}_A}$, there is always some unitary operation${\displaystyle U:{\mathcal{H}}_A\to {\mathcal{H}}_A}$ such that$${\displaystyle |{\mathrm{\Psi }}^{\prime }⟩=(I\otimes U)|\mathrm{\Psi }⟩.}$$

TheSchrödinger–HJW theorem is a result about the realization of amixed state of aquantum system as anensemble ofpure quantum states and the relation between the corresponding purifications of thedensity operators. The theorem is named afterErwin Schrödinger who proved it in 1936,^[3] and afterLane P. Hughston,Richard Jozsa andWilliam Wootters who rediscovered in 1993.^[4] The result was also found independently (albeit partially) byNicolas Gisin in 1989,^[5] and by Nicolas Hadjisavvas building upon work byE. T. Jaynes of 1957,^[6][7] while a significant part of it was likewise independently discovered byN. David Mermin in 1999 who discovered the link with Schrödinger's work.^[8] Thanks to its complicated history, it is also known by various other names such as theGHJW theorem,^[9] theHJW theorem, and thepurification theorem.

Consider a mixed quantum state${\displaystyle \rho }$ with two different realizations as ensemble of pure states as$\rho =\sum _i{p}_i|{\varphi }_i⟩⟨{\varphi }_i|$ and$\rho =\sum _j{q}_j|{\phi }_j⟩⟨{\phi }_j|$. Here both${\displaystyle |{\varphi }_i⟩}$and${\displaystyle |{\phi }_j⟩}$ are not assumed to be mutually orthogonal. There will be two corresponding purifications of the mixed state${\displaystyle \rho }$ reading as follows:

Purification 1:${\displaystyle |{\mathrm{\Psi }}_{SA}^1⟩=\sum _i\sqrt{p_i}|{\varphi }_i⟩\otimes |a_i⟩}$;

Purification 2:${\displaystyle |{\mathrm{\Psi }}_{SA}^2⟩=\sum _j\sqrt{q_j}|{\phi }_j⟩\otimes |b_j⟩}$.

The sets${\displaystyle \{|a_i⟩\}}$and${\displaystyle \{|b_j⟩\}}$ are two collections of orthonormal bases of the respective auxiliary spaces. These two purifications only differ by a unitary transformation acting on the auxiliary space, namely, there exists a unitary matrix${\displaystyle U_A}$ such that${\displaystyle |{\mathrm{\Psi }}_{SA}^1⟩=(I\otimes U_A)|{\mathrm{\Psi }}_{SA}^2⟩}$.^[10] Therefore,$|{\mathrm{\Psi }}_{SA}^1⟩=\sum _j\sqrt{q_j}|{\phi }_j⟩\otimes U_A|b_j⟩$, which means that we can realize the different ensembles of a mixed state just by making different measurements on the purifying system.

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